| L(s) = 1 | − 1.41·5-s − 3·9-s + 1.41·13-s + 7.07·17-s − 2.99·25-s − 4·29-s − 12·37-s − 12.7·41-s + 4.24·45-s − 14·53-s − 15.5·61-s − 2.00·65-s + 15.5·73-s + 9·81-s − 10.0·85-s − 4.24·89-s − 7.07·97-s + 12.7·101-s − 20·109-s + 14·113-s − 4.24·117-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 9-s + 0.392·13-s + 1.71·17-s − 0.599·25-s − 0.742·29-s − 1.97·37-s − 1.98·41-s + 0.632·45-s − 1.92·53-s − 1.99·61-s − 0.248·65-s + 1.82·73-s + 81-s − 1.08·85-s − 0.449·89-s − 0.717·97-s + 1.26·101-s − 1.91·109-s + 1.31·113-s − 0.392·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.940696255799733527976757968662, −8.144234687068523397466885463802, −7.64924418530679762834518186282, −6.58910494162729178610576965446, −5.67525630074849547285919795819, −4.99938843313010453410834149148, −3.63974803479058916205388777857, −3.19073958456411393365617171216, −1.64682263510376091891555248700, 0,
1.64682263510376091891555248700, 3.19073958456411393365617171216, 3.63974803479058916205388777857, 4.99938843313010453410834149148, 5.67525630074849547285919795819, 6.58910494162729178610576965446, 7.64924418530679762834518186282, 8.144234687068523397466885463802, 8.940696255799733527976757968662
